Mathematical Sciences: Topics in Ergodic Theory and OperatorTheory

Solomyak将研究与遍历理论相关的算子理论中的几个主题。第一个与K-算子代数理论有关，并以作用于Bratelli图的路径上的Vershik的所谓“进变换”为中心。这些变换定义了保测度系统，我们将研究其遍历性和谱性。该方法涉及维群、测度的调和分析和实数的特殊位数展开。这些扩展将在项目的第二部分进行考虑。第三个主题涉及确定Volterra卷积算子的不变子空间结构的问题。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类时与几何联系在一起。

Solomyak will investigate several topics in operator theory related to ergodic theory. The first has connections with K- theory of operator algebras, and centers on the so-called "adic transformations" of Vershik acting on paths of a Bratelli diagram. These transformations define measure-preserving systems whose ergodic and spectral properties will be studied. The proposed approach involves dimension groups, harmonic analysis of measures, and special digit expansions of real numbers. These expansions will be considered in the second part of the project. The third topic concerns the problem of determining the invariant subspace structure for Volterra convolution operators. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.